Given a regular or uniform tessellation of hyperbolic plane, is there a way to find a group of cells that will tile the whole plane?

For example: in the $(6,6,7)$ tessellation (truncated $\{3,7\}$), the "ratio" of hexagons to heptagons is $7:3$ (as each vertex can be understood as $2\cdot 1/6$ of hexagon + $1/7$ of heptagon). A group of $7$ hexagons and $3$ heptagons could be theoretically a reptile (and any reptile must have a multiple of these numbers).

I know of one shape in this tessellation that will tile the whole hyperbolic plane -- it has 40 tiles (28 hexagons and 12 heptagons) and it's used in several ingenious ways in the game HyperRogue (http://zenorogue.blogspot.cz/2015/02/hyperrogue-60-five-new-lands.html has an explanation).

I'd like to find some alternate shapes to suggest for this game, but I'm not sure whether there has been any research done in this area.

  • $\begingroup$ The meaning of your question is not clear. There is a lot of undefined jargon and notation: What is a "reptile"? What is the "$(6,6,7)$ tesselation"? What does does "truncated $\{3,7\}$" mean? And so on... $\endgroup$ – Lee Mosher Jul 21 '15 at 21:20
  • $\begingroup$ (6,6,7) means a tessellation with configuration of hexagon, hexagon and heptagon around a vertex. This is a truncated form of {3,7}, a tessellation with seven triangles around each vertex. {3,7} is Schläfli symbol and "truncated" is a term defined for uniform tilings (en.wikipedia.org/wiki/Truncated_order-7_triangular_tiling). $\endgroup$ – Marek14 Jul 22 '15 at 7:02
  • $\begingroup$ "Reptile" (I eventually realized this term is wrong here and tried to purge it from the question) is a shape whose copies can be used to assemble a larger version of itself; but that's actually impossible in hyperbolic plane -- what I mean is a shape that would tile the whole hyperbolic plane while simultaneously being composed of cells of the underlying tessellation. $\endgroup$ – Marek14 Jul 22 '15 at 7:05
  • $\begingroup$ Just an update, playing around in the game's sandbox mode I managed to find some new plane-tiling configurations with 10 and 50 cells, respectively. $\endgroup$ – Marek14 Jul 23 '15 at 7:23
  • $\begingroup$ no amount of tiles can ever tile the whole hyperbolic plane, (the hyperbolic plane is just to big for that) $\endgroup$ – Willemien Jul 23 '15 at 18:23

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